[论文解读] Polynomial Kernels with Reachability for Weighted $d$-Matroid Intersection

Chien-Chung Huang, Naonori Kakimura+2 more|arXiv (Cornell University)|Mar 18, 2026

Complexity and Algorithms in Graphs被引用 0

一句话总结

论文为带权 d-模切交集实现可达核化，为若干模类给出多项式核，并通过新的可达核技术扩展至更广场景。

ABSTRACT

This paper studies randomized polynomial kernelization for the weighted $d$-matroid intersection problem. While the problem is known to have a kernel of size $O(d^{(k - 1)d})$ where $k$ is the solution size, the existence of a polynomial kernel is not known, except for the cases when either all the given matroids are partition matroids~(i.e., the $d$-dimensional matching problem) or all the given matroids are linearly representable. The main contribution of this paper is to develop a new kernelization technique for handling general matroids. We first show that the weighted $d$-matroid intersection problem admits a polynomial kernel when one matroid is arbitrary and the other $d-1$ matroids are partition matroids. Interestingly, the obtained kernel has size $ ilde{O}(k^d)$, which matches the optimal bound~(up to logarithmic factors) for the $d$-dimensional matching problem. This approach can be adapted to the case when $d-1$ matroids in the input belong to a more general class of matroids, including graphic, cographic, and transversal matroids. We also show that the problem has a kernel of pseudo-polynomial size when given $d-1$ matroids are laminar. Our technique finds a kernel such that any feasible solution of a given instance can reach a better solution in the kernel, which is sufficiently versatile to allow us to design parameterized streaming algorithms and faster EPTASs.

研究动机与目标

• Motivate and study kernelization for the weighted d-matroid intersection problem parameterized by solution size k.
• Develop a general reachable kernel technique that works with one arbitrary matroid and various structured matroids.
• Obtain polynomial kernels for specific matroid classes and their combinations, including partition, graphic, cographic, transversal, and laminar matroids.
• Show how the reachable kernel supports extensions to streaming algorithms and faster EPTASs.

提出的方法

• Introduce the concept of a reachable kernel where any feasible solution of size at most k can be improved within the kernel via a sequence of exchanges with constant probability.
• Present a sampling-based kernel construction that achieves the required exchange property (SingleEXC) for matroid intersections.
• Prove that the constructed kernel satisfies the reachable exchange property for various matroid families using greedy exchange arguments (Algorithm 1).
• Derive kernel size bounds for different matroid classes: 3-branch bound for partition matroids; tilde-O(k^d) under g(k)-coverability; specialized bounds for graphic, cographic, transversal, and laminar matroids.
• Extend the approach to d-matchoid settings and the weighted matching problem with a matroid constraint, including deterministic reachability variants.

实验结果

研究问题

• RQ1 Can weighted d-matroid intersection admit a polynomial kernel when one matroid is arbitrary and the other d-1 matroids are simple or possess additional structure?
• RQ2 What kernel size can be achieved for various matroid classes (partition, graphic, cographic, transversal, laminar) in the weighted d-matroid intersection?
• RQ3 How can the new reachable kernel technique be adapted to streaming models and to improve EPTASs for budgeted matroid intersection?
• RQ4 What deterministic guarantees are possible for reachable kernels in special cases (e.g., 2-matroid intersection)?
• RQ5 Can the framework be extended to the d-matchoid setting and to rainbow/matroid-constrained matchings with polynomial kernels?

主要发现

• A reachable kernel of size Õ(k^d) is achievable when d−1 matroids are simple partition matroids and the remaining matroid is arbitrary.
• A reachable kernel of size Õ(k · ∏i g_i(k)) is achievable when the d−1 matroids are g_i(k)-coverable; this generalizes partition matroids and yields concrete bounds for graphic and cographic matroids.
• Corollaries show polynomial kernels for the weighted d-matroid intersection with all but one matroid graphic (or cographic), giving Õ(k^{2d−1}) and Õ(k^d) bounds respectively.
• The paper provides a reduction technique to convert transversal matroids to a form amenable to the kernel, and shows a kernel of Õ(k^d) for laminar matroids with a quasi-polynomial dependence k^{O(d log k)}.
• For the weighted matching problem with a matroid constraint, a reachable kernel of size Õ(k^3) is obtained.
• A deterministic reachable kernel is possible for the 2-matroid case with a g(k)-coverable matroid and an arbitrary matroid, achieving size k^2(g(k)+1).